Mid-concavity of survival probability for isotropic Levy processes

Let $X$ be a symmetric, pure jump, unimodal Levy process in $\mathbb{R}$ with an infinite Levy measure. We prove that for any fixed $t>0$ the survival probability $P^x(\tau_{(-a,a)}>t)$ is nondecreasing on $(-a,0]$, nonincreasing on $[0,a)$ and concave on $(-a/2,a/2)$, where $a>0$ and $\tau_{(-a,a)}$ is the first exit time of the process $X$ from $(-a,a)$. We also show a similar statement for sets $(-a,a) \times F \subset \mathbb{R}^d$.


Introduction
The main purpose of this paper is to investigate the monotonicity and concavity properties of the survival probability for some Lévy processes in R d . Let τ D = inf{t ≥ 0 : X t / ∈ D} be the first exit time of an open, nonempty set D ⊂ R d of the process X. We first formulate our result in one-dimensional setting. Theorem 1.1. Let X be a symmetric, pure jump, unimodal Lévy process in R with an infinite Lévy measure. Let D = (−a, a), where a > 0. Put ψ D t (x) = P x (τ D > t) for t ≥ 0 and x ∈ R. Then for any t > 0 the function x → ψ D t (x) is nondecreasing on (−a, 0], nonincreasing on [0, a) and concave on (−a/2, a/2).
In Section 4 we will apply these results to obtain analogous properties of first eigenfunctions for the related Dirichlet eigenvalue problem.
The above results for isotropic α-stable processes in R d (where α ∈ (0, 2]) and intervals (−a, a) or hyperrectangles d i=1 (−a i , a i ) are well known. They were proved by R. Bañuelos, T. Kulczycki and P. Méndez-Hernández in [2]. Indeed, the methods from [2] allow to extend these results for intervals or hyperrectangles to arbitrary subordinated Brownian motions in R d .
The main novelty of the results in this paper is that they concern arbitrary isotropic pure jump, unimodal Lévy processes in R d with an infinite Lévy measure. The method used in the proof of Theorems 1.1, 1.2 is completely different than the method used in [2]. The key idea of the proof of Theorems 1.1, 1.2 is probabilistic. A very important step in this proof is the use of some results concerning the so-called difference processes which were introduced in [8].
The proof in [2] is analytical. The main idea in [2] (for D = (−a, a)) is to prove some properties of for gaussian kernels p t (x) and then use subordination to show monotonicity and midconcavity for x → P x (X t 1 ∈ D, . . . , X tn ∈ D) (1) The results for P x (τ D > t) in [2] follows by a limiting procedure.
Note that in this paper we do not study properties of the function (1) but we study only properties of P x (τ D > t).
Very recently many researchers have been studying convexity properties of solutions of equations involving fractional Laplacians see [1], [4], [6], [7], [10]. In particular, concavity properties of the first eigenfunction for the Dirichlet eigenvalue problem on an interval for the fractional Laplacians have been studied in [1] and [6]. In this paper, using a probalistic approach, we obtain concavity properties of the first eigenfunction for the Dirichlet eigenvalue problem on an interval for much more general nonlocal operators, namely generators of the isotropic unimodal Lévy processes.
The paper is organized as follows. In Section 2 we present notation and collect some known facts needed in the rest of the paper. Section 3 contains proofs of Theorems 1.1, 1.2. In Section 4 we present regularity results of first eigenfunctions for the related Dirichlet eigenvalue problem.

Preliminaries
For x ∈ R d and r > 0 we let B(x, r) = {y ∈ R d : |y − x| < r}. A Borel measure on R d is called isotropic unimodal if on R d \ {0} it is absolutely continuous with respect to the Lebesgue measure and has a finite radial, radially nonincreasing density function (such measures may have an atom at the origin).
A Lévy process X = (X t , t ≥ 0) in R d is called isotropic unimodal if its transition probability p t (dx) is isotropic unimodal for all t > 0. When additionally X is a pure-jump process then the following Lévy-Khintchine formula holds for t > 0 and ξ ∈ R d , ψ is the characteristic exponent of X and ν is the Lévy measure of X. E 0 is the expected value for the process X starting from 0. Recall that a Lévy measure is a measure concentrated on R d \ {0} such that R d (|x| 2 ∧ 1)ν(dx) < ∞. Isotropic unimodal pure-jump Lévy measures are characterized in [12] by unimodal Lévy Unless explicitly stated otherwise in what follows we assume that X is a purejump isotropic unimodal Lévy process in R d with (isotropic unimodal) infinite Lévy measure ν. Then for any t > 0 the measure p t (dx) has a radial, radially nonincreasing density function p t (x) = p t (|x|) on R d with no atom at the origin. However, it may happen that p t (0) = ∞, for some t > 0. As usual, we denote by P x and E x the probability measure and the corresponding expectation for the the process starting from x ∈ R d .
Let D ⊂ R d be an open, nonempty set. We define a killed process is called the harmonic measure with respect to X. The harmonic measure for Borel sets A ⊂ (D) c is given by the Ikeda-Watanabe formula [5], When D ⊂ R d is a bounded, open Lipschitz set then we have [11], [9], It follows that for such sets D the Ikeda-Watanabe formula (3) By the Ikeda-Watanabe formula [5] for any Borel A ⊂ (0, ∞), B ⊂ (D) c we have If (4) holds then we can take B ⊂ D c in (6).

The monotonicity and midconcavity
We will prove both Theorems 1.1, 1.2 simultaneously. Let X be an isotropic, pure jump, unimodal Lévy process in R d , d ≥ 1 with an infinite Lévy measure. Let D = (−a, a) × F , where a > 0 and F ⊂ R d−1 is a bounded Lipschitz domain (or D = (−a, a) when d = 1). Put e 1 = (1, 0 . . . , 0) ∈ R d . Note that for any x ∈ D we have P x (X(τ D ) ∈ ∂D) = 0.
The key point in this section is the following result.
For any x ∈ U + , t > 0 we have Proof. By the strong Markov property and (6) for any x ∈ U, t > 0 we have It follows that For any x ∈ U + , t > 0 by the symmetry of the process X and the definition of For any x ∈ U + , t > 0, z ∈ (U ) c we also have U (p U (s, x, y) − p U (s, T U (x), y)) ν(y − z) dy (where in the last equality we used p U (s, T U (x), T U (y)) = p U (s, x, y)). Applying this, (10)(11) and (12) we get (7).
Note that p U (s, x, y) − p U (s, T U (x), y) is the transition density of the so-called difference process (with respect to the hyperplane x 1 = m(U) or the point m(u) when d = 1) killed on exiting U + (see Section 4 in [8] for more details). By (19) in [8] and the first formula after the proof of Lemma 4.3 in [8] we obtain that p U (s, x, y) − p U (s, T U (x), y) ≥ 0 for any s > 0, x, y ∈ U + . By unimodality of ν(x) we obtain that ν(y − z) − ν (T U (y) − z) ≥ 0 for any y ∈ U + , z ∈ H + (U) \ U + and ν(y − z) − ν (T U (y) − z) ≤ 0 for any y ∈ U + , z ∈ H − (U) \ U − . This gives (8) and (9). Now we will show our main results.
proof of Theorems 1.1, 1.2. First we study monotonicity of (7) for any t > 0 we get This, (13-15) and (8) give . It follows that the function y → ψ D t (ye 1 +x) (or y → ψ D t (y) when d = 1) is nondecreasing on (−a, 0]. By symmetry of the process X and the domain D the function y → ψ D t (ye 1 +x) (or y → ψ D t (y) when d = 1) is nonincreasing on [0, a). Now we will study midconcavity of the function ψ D t . Fixx ∈ {0} × F and −a/2 < x ′ Using this and Proposition 3.1 applied to W we get for any t > 0 Note that W + = U + + v and x ′′′ = x ′′ + v. Using this and the definition of f W s (x, z) we get for any Using substitution q = y − v this is equal to For any s > 0, q ∈ U + we have By the definition of T W and the equality m(W ) = m(U) + v 1 we get Hence for any s > 0, q ∈ U + we obtain By similar arguments as above for any q ∈ U + we get T W (q + v) = T U (q) + v. Hence for any q ∈ U + and z ∈ W c we obtain Using this, (18-21) and (22-23) we get for any s > 0, z ∈ (W ) c . Using this, the fact that W c = U c + v and (16-17) we get for any t > 0 Put such that for any n ∈ N, t > 0, x ∈ D we have P D t ϕ n (x) = e −λnt ϕ n (x).
λ 1 has multiplicity one and we may assume that ϕ 1 > 0 on D. By properties of p D (t, x, y) all eigenfunctions ϕ n are bounded and continuous on D. It is well known that p D (t, x, y) = ∞ n=1 e −λnt ϕ n (x)ϕ n (y), t > 0, x, y ∈ D.
It follows that for any t > 0 and x ∈ D we have P x (τ D > t) = Hence for any x ∈ D we have lim t→∞ e λ 1 t P x (τ D > t) = ϕ 1 (x) D ϕ 1 (y) dy.
Using this and Theorems 1.1, 1.2 we immediately obtain the following results.